# Positive multi-peak solutions for a logarithmic Schrodinger equation

## TL;DR

This paper proves the existence and local uniqueness of positive multi-peak solutions for a singularly perturbed logarithmic Schrödinger equation using Lyapunov-Schmidt reduction and a novel norm to handle the nonlinearity.

## Contribution

It introduces a new analytical approach to study multi-peak solutions for the logarithmic Schrödinger equation, a problem not previously addressed with reduction techniques.

## Key findings

- Existence of positive multi-peak solutions under certain conditions on V(x)
- Development of a new norm to manage the logarithmic nonlinearity
- Establishment of local uniqueness of solutions using Pohozaev identities

## Abstract

In this manuscript, we consider the logarithmic Schr\"{o}dinger equation \begin{eqnarray*} -\varepsilon^2\Delta u+V(x)u=u\log u^{2},\,\,\,u>0, & \text{in}\,\,\,\mathbb{R}^{N}, \end{eqnarray*} where $N\geq3$, $\varepsilon>0$ is a small parameter. Under some assumptions on $V(x)$, we show the existence of positive multi-peak solutions by Lyapunov-Schmidt reduction. It seems to be the first time to study singularly perturbed logarithmic Schr\"{o}dinger problem by reduction. And here using a new norm is the crucial technique to overcome the difficulty caused by the logarithmic nonlinearity. At the same time, we consider the local uniqueness of the multi-peak solutions by using a type of local Pohozaev identities.